McKay, BrendanTian, Fang2023-12-110196-8858http://hdl.handle.net/1885/309789For n ≥ 3, let r = r(n) ≥ 3 be an integer. A hypergraph is r-uniform if each edge is a set of r vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear r-uniform hypergraphs on n → ∞ vertices is determined asymptotically when the number of edges is m(n) = o(r−3n3/2). As one application, we find the probability of linearity for the independent-edge model of random r-uniform hypergraph when the expected number of edges is o(r−3n3/2). We also find the probability that a random r-uniform linear hypergraph with a given number of edges contains a given subhypergraph.Fang Tian was partially supported by the National Natural Science Foundation of China (Grant No. 11871377) and China Scholarship Council [2017]3192, and is now a visiting research fellow at the Australian National University. Fang Tian is immensely grateful to Brendan D. McKay for giving her the opportunity to learn from him, and thanks him for his problem and useful discussions.application/pdfen-AU© 2020 The authorsAsymptotic enumerationLinear hypergraphSwitching methodRandom hypergraph202010.1016/j.aam.2020.1020002022-09-04